Arithmetic Concepts in GMAT Quant

A trick to tackling GMAT probability questions

The concept of probability is a simple one, yet its application often trips up GMAT test-takers.

Basic formula for calculating probability:

Probability = number of wanted outcomes / total number of outcomes

In essence, probability, like a percentage, is a ratio between a part and a whole, expressed as a fraction.

Now see if you can apply it to this sample from our bank of 5000+ practice questions:

A canoe has two oars, left and right. Each oar either works or breaks. The failure or non-failure of each oar is independent of the failure or non-failure of the other. You can still row the canoe with one oar. The probability that the left oar works is 3/5. The probability that the right oar works is also 3/5. What is the probability that you can still row the canoe?

A. 9/25B.10/25C. 6/10D. 2/3E. 21/25

1. The wrong way to solve this probability problem

The temptation for our students is often to multiply the two probabilities given to reach the answer 9/25. If you did this, keep the following tip in mind:

Whenever you get to an answer choice very quickly, particularly when that answer is A, consider looking at the question again! Answer choice A is the first answer you see. If you are in a hurry and option A looks right, many test takers will go for A.

Here's why it's wrong:

This calculation only gives you the probability that both oars work.

To get the right answer, you would also have to add the probability that the left oar works and the right fails.

Then you would also have to add the probability that the right works, but the left fails.

All this is 100% possible, but slow. One of the things we emphasize throughout the entirety of our GMAT Tutor course is not just solving problems correctly, but solving them in the fastest way possible. Is there a better way? Yes!

2. The right way to solve this probability problem

Simply look at the question from the other side. What is the probability that you can’t row the canoe? This would be 2/5 x 2/5 = 4/25.

Using the idea that the probability of anything happening is 1 (a.k.a. 100%) minus the probability that it doesn’t happen, you can use the following equation to reach the right answer:

1 – 4/25 = 21/25

In other words, answer choice E.

At the Economist GMAT Tutor, we call this the Forbidden Method: subtracting the ‘forbidden’ or unwanted probabilities from the total probability, which is 1.

Let's try another application of the probability formula.

Applying the probability formula to a tricky word problem:

Set your timer for two minutes, and see if you can get the correct answer. Then read below for the full explanation, and see if you’ve fallen into a trap or not!

Sixty percent of the members of a study group are women, and 45 percent of those women are lawyers. If one member of the study group is to be selected at random, what is the probability that the member selected is a woman lawyer?

(A) 0.10(B) 0.15(C) 0.27(D) 0.33(E) 0.45

Although the question uses the word “probability,” the concept it tests for and the trap laid within are percent-related – the piece of information you need from the wide field of probability to solve this problem is the basic formula:

Probability = number of wanted outcomes / total number of outcomes

This problem is deceptively easy, which is why a decent percent of GMAT test takers will get it wrong, at least when put under a time crunch. For many test takers, the following (mistaken) thought pattern will ensue:

60% of the members are women. Imagine a pie chart, with a 60% chunk marked “women.”Now, 45% are lawyers. Take a chunk of 45% (almost half of the pie), out of the original 60% chunk, and that’s your percent of women lawyers – 45%, or 0.45 (answer choice E).

In extreme rush cases, a test-taker may even forget what he or she is looking for in the first place. Once you’re imagining a 45% chunk taken out of the 60% chunk, it’s deceptively easy to fall into the trap of focusing on what remains: a 15% “slice,” which while hurrying will lead you into choosing B in the rush to move on to the next question.

Both of these thought processes and the resulting answer choices are wrong.

The stumbling point that the GMAT test-writers are counting on is the failure to ask these simple questions whenever the word “percent” appears: What is the whole?What number or quantity is the percent taken out of?

The first percent (60% women) is indeed taken out of the members of the study group. The next line has a crucial phrase: 45% of those women are lawyers. So the next percent is not taken out of the entire pie chart, but out of the 65% chunk alone. We’re looking for 45% of the group titled women, which happens to be given as a percent of the whole, not just 45% of the entire group.

The actual calculation is therefore 45% of 65%, or 45/100 × 65/100 (think of any “of” in these cases as a multiplication sign).

One last note: instead of actually calculating the above expression, you can always just ballpark it. The group you seek (women lawyers) is ‘slightly less than half’ of the women (as 45% is just under 50%).

The right answer will therefore need to be something slightly smaller than 1/2 × 0.6 = 0.3.

Only one answer choice out of the five answer choices presented fits that description, and that is C) 0.27

Answer choices A and B are too small, and D and E are already over half of 60%.

This was a sample of the in-depth instruction that Economist GMAT Tutor offers about solving probability questions in the GMAT Quant section. For complete and interactive lessons, practice tests, and online tutor support, subscribe to one of Economist GMAT Tutor's top-rated GMAT prep plans. Commitment-free trials are available for seven days.

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